潜在结果框架

实验研究

For unit $i$ , we have pretreatment covariates $X_{i}$ , a binary treatment indicator $D_{i}$ , and an observed outcome $Y_{i}$ with two potential outcomes $Y_{i} (1)$ and $Y_{i} (0)$ under the treatment and control, respectively. For simplicity, we assume

{D_{i}, X_{i}, Y_{i} (0), Y_{i} (1)}_{i = 1}^{n} \overset{i i d}{\sim} {D, X, Y (0), Y (1)}

So we can drop the subscript $i$ for quantities of this population.

Definition Potential outcome of $Y$

Y = {\begin{cases} Y (0), if D = 0 \\ Y (1), if D = 1 \end{cases}

Definition Treatment effect of unit $i$

T E \equiv Y (0) - Y (1)

Definition Average treatment effect on the treated

A T T \equiv E [T E ∣ D = 1] = E [Y (1) - Y (0) ∣ D = 1]

Definition Average treatment effect on the untreated

A T U \equiv E [T E ∣ D = 0] = E [Y (1) - Y (0) ∣ D = 0]

Definition Average Treatment Effect

A T E \equiv E [T E] = E [Y (1) - Y (0)] = A T T \times P (D = 1) + A T U \times P (D = 0)

Definition 差分被估量(naive estimand)

Δ = E [Y ∣ D = 1] - E [Y ∣ D = 0] = E [Y (1) ∣ D = 1] - E [Y (0) ∣ D = 0]

比较 $Δ$ 和 $A T T$ 、 $A T U$ 的差异

\begin{aligned} Δ - A T T & = E [Y (0) ∣ D = 1] - E [Y (0) ∣ D = 0] \\ Δ - A T U & = E [Y (1) ∣ D = 1] - E [Y (1) ∣ D = 0] \end{aligned}

这说明如果处理组和对照组的潜在结果不同，估计组平均处理效应时就会存在选择偏差。

比较 $Δ$ 和 $A T E$ 的差异

\begin{aligned} Δ - A T E & = Δ - A T T \times P (D = 1) - A T U \times P (D = 0) \\ = (Δ - A T T) + (A T T - A T U) \times P (D = 0) \\ = (Δ - A T U) + (A T U - A T T) \times P (D = 1) \end{aligned}

两个等式对应两种分解方法，分解为相应组的选择偏差和异质处理效应。

引入独立性假设（Independence Assuption, IA）

{Y (0), Y (1)} ⊥ ⊥ D

即是否接受处理是随机的，则有

Δ = A T T = A T U = A T E

此时，选择偏差和异质处理效应全部消失。

观察研究

Definition Conditional average treatment effect on the treated

A T T (X) \equiv E [Y (1) - Y (0) ∣ D = 1, X]

Definition Conditional average treatment effect on the untreated

A T U (X) \equiv E [Y (1) - Y (0) ∣ D = 0, X]

Definition Conditional average Treatment Effect

A T E (X) \equiv E [Y (1) - Y (0) ∣ X] = A T T (X) \times P (D = 1 ∣ X) + A T U (X) \times P (D = 0 ∣ X)

Definition 条件差分被估量(naive estimand)

Δ (X) = E [Y ∣ D = 1, X] - E [Y ∣ D = 0, X] = E [Y (1) ∣ D = 1, X] - E [Y (0) ∣ D = 0, X]

比较 $Δ (X)$ 和 $A T T (X)$ 、 $A T U (X)$ 的差异

\begin{aligned} Δ (X) - A T T (X) & = E [Y (0) ∣ D = 1, X] - E [Y (0) ∣ D = 0, X] \\ Δ (X) - A T U (X) & = E [Y (1) ∣ D = 1, X] - E [Y (1) ∣ D = 0, X] \end{aligned}

比较 $Δ (X)$ 和 $A T E (X)$ 的差异

\begin{aligned} Δ (X) - A T E (X) & = Δ (X) - A T T (X) \times P (D = 1 ∣ X) - A T U (X) \times P (D = 0 ∣ X) \\ = [Δ (X) - A T T (X)] + [A T T (X) - A T U (X)] \times P (D = 0 ∣ X) \\ = [Δ (X) - A T U (X)] + [A T U (X) - A T T (X)] \times P (D = 1 ∣ X) \end{aligned}

引入条件独立性假设（Conditional Independence Assuption, CIA）

{Y (0), Y (1)} ⊥ ⊥ D ∣ X

即给定 $X$ 后是否接受处理是随机的，则有

Δ (X) = A T T (X) = A T U (X) = A T E (X)

此时，选择偏差和异质处理效应全部消失。

根据定义易知

\begin{aligned} A T T & = \sum A T T (X = x) P (X = x ∣ D = 1) \\ A T U & = \sum A T U (X = x) P (X = x ∣ D = 0) \\ A T E & = \sum A T E (X = x) P (X = x) \end{aligned}

因此，在 CIA 假设下使用 $Δ (X)$ 即可估计任意因果效应。

Attention

条件差分被估量 $Δ (X)$ 的存在要求 $X$ 划分任意的层同时包含处理组和对照组，这被称为共同支撑条件(common support condition)，记作 $0 < P (D = 1 ∣ X) < 1$ 。事实上，共同支撑条件在 $X$ 维数很高时难以满足，这为倾向得分的出现埋下了伏笔。